Optimal. Leaf size=46 \[ \frac{\text{Erf}(b x)}{4 b^2}-\frac{x e^{-b^2 x^2}}{2 \sqrt{\pi } b}+\frac{1}{2} x^2 \text{Erfc}(b x) \]
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Rubi [A] time = 0.0346855, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6362, 2212, 2205} \[ \frac{\text{Erf}(b x)}{4 b^2}-\frac{x e^{-b^2 x^2}}{2 \sqrt{\pi } b}+\frac{1}{2} x^2 \text{Erfc}(b x) \]
Antiderivative was successfully verified.
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Rule 6362
Rule 2212
Rule 2205
Rubi steps
\begin{align*} \int x \text{erfc}(b x) \, dx &=\frac{1}{2} x^2 \text{erfc}(b x)+\frac{b \int e^{-b^2 x^2} x^2 \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{-b^2 x^2} x}{2 b \sqrt{\pi }}+\frac{1}{2} x^2 \text{erfc}(b x)+\frac{\int e^{-b^2 x^2} \, dx}{2 b \sqrt{\pi }}\\ &=-\frac{e^{-b^2 x^2} x}{2 b \sqrt{\pi }}+\frac{\text{erf}(b x)}{4 b^2}+\frac{1}{2} x^2 \text{erfc}(b x)\\ \end{align*}
Mathematica [A] time = 0.0417538, size = 43, normalized size = 0.93 \[ \frac{1}{4} \left (\frac{\text{Erf}(b x)}{b^2}+2 x \left (x \text{Erfc}(b x)-\frac{e^{-b^2 x^2}}{\sqrt{\pi } b}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 46, normalized size = 1. \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{b}^{2}{x}^{2}{\it erfc} \left ( bx \right ) }{2}}+{\frac{1}{\sqrt{\pi }} \left ( -{\frac{bx}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}+{\frac{\sqrt{\pi }{\it Erf} \left ( bx \right ) }{4}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14194, size = 59, normalized size = 1.28 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{erfc}\left (b x\right ) - \frac{b{\left (\frac{2 \, x e^{\left (-b^{2} x^{2}\right )}}{b^{2}} - \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )}{b^{3}}\right )}}{4 \, \sqrt{\pi }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53695, size = 122, normalized size = 2.65 \begin{align*} \frac{2 \, \pi b^{2} x^{2} - 2 \, \sqrt{\pi } b x e^{\left (-b^{2} x^{2}\right )} +{\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right )}{4 \, \pi b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.448672, size = 42, normalized size = 0.91 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{erfc}{\left (b x \right )}}{2} - \frac{x e^{- b^{2} x^{2}}}{2 \sqrt{\pi } b} - \frac{\operatorname{erfc}{\left (b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32194, size = 66, normalized size = 1.43 \begin{align*} -\frac{1}{2} \, x^{2} \operatorname{erf}\left (b x\right ) + \frac{1}{2} \, x^{2} - \frac{b{\left (\frac{2 \, x e^{\left (-b^{2} x^{2}\right )}}{b^{2}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-b x\right )}{b^{3}}\right )}}{4 \, \sqrt{\pi }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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